NeuralDynamics_KuramotoInteractive.ipynb

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   "source": [
    "# Neural Network Conceptualization using Kuramoto Model\n",
    "\n",
    "This notebook explores the Kuramoto model, a complex system representing synchronization, and draws analogies to neural network behaviors. By adjusting the model parameters, we can gain insights into how neural networks might behave under varying conditions.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Create a venv with python i.e. \n",
    "\n",
    "conda create -n fractals python=3.12.1\n",
    "\n",
    "conda activate fractals\n",
    "\n",
    "# Make sure you have the right kernel selected in the top right of VS Code.\n",
    "\n",
    "*You may need to:* \n",
    "\n",
    "pip install matplotlib\n",
    "\n",
    "pip install ipywidgets\n",
    "\n",
    "*Then you can run the rest*-"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from matplotlib.animation import FuncAnimation\n",
    "import ipywidgets as widgets\n",
    "from IPython.display import display"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Interactive Controls\n",
    "\n",
    "Use the sliders below to adjust the parameters of the Kuramoto model. These parameters can be thought of as analogous to aspects of a neural network:\n",
    "- **Number of Oscillators**: Similar to the number of neurons.\n",
    "- **Coupling Strength**: Analogous to the strength of synaptic connections.\n",
    "- **Noise Level**: Represents external inputs or perturbations to the system.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Widget definitions\n",
    "n_oscillators_widget = widgets.IntSlider(min=10, max=1000, step=1, value=50, description='Oscillators')\n",
    "coupling_strength_widget = widgets.FloatSlider(min=0, max=200, step=0.1, value=0.5, description='Coupling Strength')\n",
    "noise_level_widget = widgets.FloatSlider(min=0, max=100, step=0.05, value=0.1, description='Noise Level')\n",
    "frequency_dist_widget = widgets.FloatSlider(min=0.001, max=200, step=0.1, value=1, description='Freq. Distribution')\n",
    "time_widget = widgets.IntSlider(min=1, max=5000, step=1, value=50, description='Time')\n",
    "phases_widget = widgets.IntSlider(min=1, max=5000, step=1, value=50, description='Phases')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Kuramoto model function \n",
    "def kuramoto_model(n, coupling_strength, frequency_dist, time, dt, noise_level):\n",
    "        phases = np.random.uniform(0, 2 * np.pi, n)  # Initial phases with applied phase shifts\n",
    "        # Natural frequencies\n",
    "        frequencies = np.random.normal(0, frequency_dist, n)\n",
    "        history = []\n",
    "        coupling_strength = 0.1  # Initial coupling strength\n",
    "    \n",
    "        for t in np.arange(0, time, dt):\n",
    "            # Varying coupling strength over time\n",
    "            coupling_strength = 0.5 + 0.5 * np.sin(2 * np.pi * t / time)\n",
    "            # Add noise to phases\n",
    "            phases += np.random.normal(0, noise_level, n) * dt\n",
    "            # Calculate phase differences matrix\n",
    "            phase_diff_matrix = np.sin(np.subtract.outer(phases, phases))\n",
    "            # Update phases\n",
    "            dtheta = dt * (frequencies + (coupling_strength / n) * np.sum(phase_diff_matrix, axis=1))\n",
    "            phases = (phases + dtheta) % (2 * np.pi)\n",
    "            history.append(phases.copy())\n",
    "    \n",
    "        return np.array(history)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Visualization function for the Kuramoto model \n",
    "def visualize_dynamic_heatmap(history):\n",
    "    fig, ax = plt.subplots()\n",
    "    im = ax.imshow(np.sin(np.subtract.outer(history[0], history[0])), cmap='hsv', interpolation='nearest')\n",
    "    fig.colorbar(im, ax=ax, orientation='vertical')\n",
    "    ax.set_title('Enhanced Kuramoto Model')\n",
    "\n",
    "    def animate(i):\n",
    "        im.set_array(np.sin(np.subtract.outer(history[i], history[i])))\n",
    "        return [im]\n",
    "\n",
    "    ani = FuncAnimation(fig, animate, frames=len(history), interval=50, blit=True)\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Neural Network Analogy\n",
    "\n",
    "The Kuramoto model's synchronization patterns can be seen as a metaphor for the behavior of neural networks. By observing the changes in synchronization, we can conceptualize how neural networks might adapt and evolve during learning processes.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
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     "data": {
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       "model_id": "a038132e309c4d9497db5c629c1bcda5",
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      "text/plain": [
       "interactive(children=(IntSlider(value=50, description='Oscillators', max=1000, min=10), FloatSlider(value=0.5,…"
      ]
     },
     "metadata": {},
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   ],
   "source": [
    "# Function to update and visualize the model based on widget values\n",
    "def update_and_visualize(n, coupling_strength, frequency_dist, time, phases, noise_level):\n",
    "    history = kuramoto_model(n, coupling_strength, frequency_dist, time, phases, noise_level)\n",
    "    visualize_dynamic_heatmap(history)\n",
    "\n",
    "# Link widgets to the visualization function\n",
    "interactive_plot = widgets.interactive(update_and_visualize, \n",
    "                                       n=n_oscillators_widget, \n",
    "                                       time=time_widget,\n",
    "                                       phases=phases_widget,\n",
    "                                       coupling_strength=coupling_strength_widget, \n",
    "                                       frequency_dist=frequency_dist_widget, \n",
    "                                       noise_level=noise_level_widget)\n",
    "\n",
    "# Display the interactive plot\n",
    "display(interactive_plot)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Discussion\n",
    "\n",
    "The interactive exploration of the Kuramoto model allows us to draw parallels with neural network training. For instance, the synchronization process can be likened to the learning phase of neural networks, where neurons adjust their weights (synaptic strengths) to achieve a specific task.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Conclusion\n",
    "\n",
    "This notebook serves as a conceptual bridge between the physics of synchronization and neural network dynamics, offering an intuitive way to understand complex behaviors in neural networks.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The toy model you've provided simulates the Kuramoto model, which is a mathematical model to describe synchronization. It represents a system of coupled oscillators with a given natural frequency and interact with each other through a sinusoidal coupling.\n",
    "\n",
    "Here are a few potential enhancements and ideas to make your model more functional for applications in Machine Learning (ML) and Artificial Intelligence (AI):\n",
    "\n",
    "1. **Data Collection and Labeling**: Record the system's state at various points in time to create a dataset that can be used for ML. Label the data based on the behavior you observe, like synchronized or desynchronized states.\n",
    "\n",
    "2. **Feature Engineering**: Develop features that could be predictive of the system's behavior. This could include statistical properties of the phases or frequencies, measures of the system's order parameter (like the degree of synchronization), or the variance in the phase difference matrix over time.\n",
    "\n",
    "3. **Predictive Modeling**: Use the data to train a machine learning model to predict future states of the system based on its current state. This could be a regression problem if predicting the exact phase, or a classification problem if predicting the behavior (synchronized or not).\n",
    "\n",
    "4. **Anomaly Detection**: Implement an anomaly detection algorithm that identifies when the system behaves in an unexpected way, which could be crucial for systems where synchronization is critical.\n",
    "\n",
    "5. **Parameter Optimization**: Use ML to optimize the parameters of the model (like the coupling strength and noise level) to achieve desired behaviors. Reinforcement learning could be an interesting approach here, where an agent learns to tune the parameters to minimize a cost function associated with undesired behavior.\n",
    "\n",
    "6. **Complex Network Analysis**: If you model a network of Kuramoto oscillators with different coupling strengths and topology, you could apply ML to identify key features that predict synchronization in complex networks.\n",
    "\n",
    "7. **Time Series Forecasting**: Given that the model evolves over time, techniques from time series analysis and forecasting could be applied to predict the evolution of the phases.\n",
    "\n",
    "8. **Control Systems**: Integrate the model with a control algorithm that could adjust the coupling strength in real-time to maintain synchronization, using approaches like PID control or more advanced ML-based control systems.\n",
    "\n",
    "9. **Robustness Testing**: Implement tests to see how the system responds to perturbations or changes in initial conditions, which can be an interesting study in resilience and robustness of synchronized systems.\n",
    "\n",
    "10. **Visualization**: Enhance the visualization of the system's dynamics over time. A dynamic, interactive visualization could provide insights into the behavior of the system that static plots do not show.\n",
    "\n",
    "For specific improvements to your current Python function, consider the following:\n",
    "\n",
    "- **Parameterize the Initial Coupling Strength**: You have a hard-coded initial coupling strength value. Make this a parameter that can be passed to the function.\n",
    "- **Variable Noise Implementation**: Instead of adding noise uniformly at every timestep, consider a model where noise can have different characteristics, e.g., more noise as the system evolves or noise that depends on the state of the system.\n",
    "- **Saving State for Restart**: Implement functionality to save and load the state of the system so you can stop and restart simulations without losing progress.\n",
    "- **Efficient Computation**: Optimize the computation of the phase difference matrix and the update of phases. Utilize efficient numpy operations to speed up these calculations, especially for larger numbers of oscillators.\n",
    "\n",
    "Once you've refined your model, you can use it as a simulation tool to generate data for ML applications or as a component of a larger AI system that interacts with complex dynamical systems."
   ]
  }
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